Existence and Uniqueness Conditions for the Maximum Likelihood Solution In Regression Models For Correlated Bernoulli Random Variables

We give sufficient and necessary conditions for the existence of the maximum likelihood estimate in a class of multivariate regression models for correlated Bernoulli random variables. The models use the concept of threshold crossing technique of an underlying multivariate latent variable with univariate components formulated as a linear regression model. However, in place of their Gaussian assumptions, any specified distribution with a strictly increasing cumulative distribution function is allowed for error terms. A well known member of this class of models is the multivariate probit model. We show that our results are a generalization of the concepts of separation and overlap of Albert and Anderson for the study of the existence of maximum likelihood estimate in generalized linear models. Implications of our findings are illustrated through some hypothetical examples.